Wiedemann-Franz law
Encyclopedia
In physics
, the Wiedemann–Franz law states that the ratio of the electronic contribution to the thermal conductivity
(κ) and the electrical conductivity (σ) of a metal
is proportional to the temperature
(T).
Theoretically, the proportionality constant L, known as the Lorenz number, is equal to
This empirical
law is named after Gustav Wiedemann
and Rudolph Franz, who in 1853 reported that κ/σ has approximately the same value for different metals at the same temperature. The proportionality of κ/σ with temperature was discovered by Ludvig Lorenz in 1872.
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal.
The mathematical expression of the law can be derived as following.
Electrical conduction of metals is a well known phenomenon and is attributed to the rather free conduction electrons. It is measured as sketched in the figure. The current density
j is observed to be proportional to the applied electric field
and follows Ohm's law
where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors we have expressed Ohm's law here in bold face. The conductivity can in general be expressed as a tensor
of the second rank (3×3 matrix
). Here we restrict the discussion to isotropic, i.e. scalar
conductivity. The specific resistivity
is the inverse of the conductivity. Both parameters will be used in the following.
Drude
(c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
The assumption is that the electrons move freely in the solid like in an ideal gas
. The force applied to the electron by the electric field leads to an acceleration
according to
This would lead, however, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight. This establishes an average or drift velocity
Vd. The drift velocity is related to the average scattering time as becomes evident from the following relations.
Kittel gives some values of L ranging from L = 2.23×10−8 W Ω K−2 for copper at 0 °C to L = 3.2×10−8 W Ω K−2 for tungsten at 100 °C. Rosenberg notes that the Wiedemann–Franz law is generally valid for high temperatures and for low (i.e., a few kelvins) temperatures, but may not hold at intermediate temperatures.
In degenerate semiconductors, the Lorenz number L has a strong dependency on certain system parameters: dimensionality, strength of interatomic interactions and Fermi-level. This law is not valid or the value of the Lorentz
number can be reduced at least in following cases: manipulating electronic density of states, varying doping density and layer thickness in superlattices and materials with correlated carriers.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the Wiedemann–Franz law states that the ratio of the electronic contribution to the thermal conductivity
Thermal conductivity
In physics, thermal conductivity, k, is the property of a material's ability to conduct heat. It appears primarily in Fourier's Law for heat conduction....
(κ) and the electrical conductivity (σ) of a metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
is proportional to the temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
(T).
Theoretically, the proportionality constant L, known as the Lorenz number, is equal to
This empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
law is named after Gustav Wiedemann
Gustav Heinrich Wiedemann
Gustav Heinrich Wiedemann was a German physicist known mostly for his literary work.-Education:Wiedemann was born at Berlin. After attending the Cologne gymnasium, he entered the university of Berlin in 1844, and took his doctor's degree there three years later...
and Rudolph Franz, who in 1853 reported that κ/σ has approximately the same value for different metals at the same temperature. The proportionality of κ/σ with temperature was discovered by Ludvig Lorenz in 1872.
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal.
The mathematical expression of the law can be derived as following.
Electrical conduction of metals is a well known phenomenon and is attributed to the rather free conduction electrons. It is measured as sketched in the figure. The current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
j is observed to be proportional to the applied electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and follows Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors we have expressed Ohm's law here in bold face. The conductivity can in general be expressed as a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
of the second rank (3×3 matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
). Here we restrict the discussion to isotropic, i.e. scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
conductivity. The specific resistivity
Resistivity
Electrical resistivity is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm metre...
is the inverse of the conductivity. Both parameters will be used in the following.
Drude
Paul Karl Ludwig Drude
Paul Karl Ludwig Drude was a German physicist specializing in optics. He wrote a fundamental textbook integrating optics with Maxwell's theories of electromagnetism.- Education :...
(c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
The assumption is that the electrons move freely in the solid like in an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
. The force applied to the electron by the electric field leads to an acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...
according to
This would lead, however, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight. This establishes an average or drift velocity
Drift velocity
The drift velocity is the average velocity that a particle, such as an electron, attains due to an electric field. It can also be referred to as Axial Drift Velocity since particles defined are assumed to be moving along a plane. In general, an electron will 'rattle around' in a conductor at the...
Vd. The drift velocity is related to the average scattering time as becomes evident from the following relations.
Limitations of the theory
Experiments have shown that the value of L, while roughly constant, is not exactly the same for all materials.Kittel gives some values of L ranging from L = 2.23×10−8 W Ω K−2 for copper at 0 °C to L = 3.2×10−8 W Ω K−2 for tungsten at 100 °C. Rosenberg notes that the Wiedemann–Franz law is generally valid for high temperatures and for low (i.e., a few kelvins) temperatures, but may not hold at intermediate temperatures.
In degenerate semiconductors, the Lorenz number L has a strong dependency on certain system parameters: dimensionality, strength of interatomic interactions and Fermi-level. This law is not valid or the value of the Lorentz
number can be reduced at least in following cases: manipulating electronic density of states, varying doping density and layer thickness in superlattices and materials with correlated carriers.